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Theory and applications of Robust Optimization
, 2007
"... In this paper we survey the primary research, both theoretical and applied, in the field of Robust Optimization (RO). Our focus will be on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying the most pr ..."
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Cited by 25 (5 self)
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In this paper we survey the primary research, both theoretical and applied, in the field of Robust Optimization (RO). Our focus will be on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying the most prominent theoretical results of RO over the past decade, we will also present some recent results linking RO to adaptable models for multistage decisionmaking problems. Finally, we will highlight successful applications of RO across a wide spectrum of domains, including, but not limited to, finance, statistics, learning, and engineering.
Selected topics in robust convex optimization
 Math. Prog. B, this issue
, 2007
"... Abstract Robust Optimization is a rapidly developing methodology for handling optimization problems affected by nonstochastic “uncertainbutbounded” data perturbations. In this paper, we overview several selected topics in this popular area, specifically, (1) recent extensions of the basic concept ..."
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Cited by 14 (2 self)
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Abstract Robust Optimization is a rapidly developing methodology for handling optimization problems affected by nonstochastic “uncertainbutbounded” data perturbations. In this paper, we overview several selected topics in this popular area, specifically, (1) recent extensions of the basic concept of robust counterpart of an optimization problem with uncertain data, (2) tractability of robust counterparts, (3) links between RO and traditional chance constrained settings of problems with stochastic data, and (4) a novel generic application of the RO methodology in Robust Linear Control. Keywords optimization under uncertainty · robust optimization · convex programming · chance constraints · robust linear control
On twostage convex chance constrained problems
, 2005
"... In this paper we develop approximation algorithms for twostage convex chance constrained problems. Nemirovski and Shapiro [18] formulated this class of problems and proposed an ellipsoidlike iterative algorithm for the special case where the impact function f(x,h) is biaffine. We show that this a ..."
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Cited by 10 (0 self)
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In this paper we develop approximation algorithms for twostage convex chance constrained problems. Nemirovski and Shapiro [18] formulated this class of problems and proposed an ellipsoidlike iterative algorithm for the special case where the impact function f(x,h) is biaffine. We show that this algorithm extends to biconvex f(x,h) in a fairly straightforward fashion. The complexity of the solution algorithm as well as the quality of its output are functions of the radius r of the largest Euclidean ball that can be inscribed in the polytope defined by a random set of linear inequalities generated by the algorithm [18]. Since the polytope determining r is random, computing r is difficult. Yet, the solution algorithm requires r as an input. In this paper we provide some guidance for selecting r. We show that the largest value of r is determined by the degree of robust feasibility of the twostage chance constrained problem – the more robust the problem, the higher one can set the parameter r. Next, we formulate ambiguous twostage chance constrained problems. In this formulation, the random variables defining the chance constraint are known to have a fixed distribution; however, the decision maker is only able to estimate this distribution to within some error. We construct an algorithm that solves the ambiguous twostage chance constrained problem when the impact function f(x,h) is biaffine and the extreme points of a certain “dual ” polytope are known explicitly. 1
Robust capacity expansion of network flows
 Networks
"... We consider the question of deciding capacity expansions for a network flow problem that is subject to demand and travel time uncertainty. We introduce a robust optimization based approach to obtain a capacity expansion solution that is insensitive to this uncertainty. We show that solving for a rob ..."
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Cited by 10 (2 self)
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We consider the question of deciding capacity expansions for a network flow problem that is subject to demand and travel time uncertainty. We introduce a robust optimization based approach to obtain a capacity expansion solution that is insensitive to this uncertainty. We show that solving for a robust solution is a computationally tractable problem for general uncertainty sets and under reasonable conditions for network flow applications. For example, the robust problem is tractable for a multicommodity flow problem with a single source and sink per commodity and uncertain demand and travel time represented by bounded convex sets. Preliminary computational results show that the robust solution is attractive, as it can reduce the worst case cost by more than 20%, while incurring on a 5 % loss in optimality when compared to the optimal solution of a representative scenario.
Computing robust basestock levels
 Discrete Optimization
"... This paper considers how to optimally set the basestock level for a single buffer when demand is uncertain, in a robust framework. We present a family of algorithms based on decomposition that scale well to problems with hundreds of time periods, and theoretical results on more general models. 1 ..."
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Cited by 10 (0 self)
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This paper considers how to optimally set the basestock level for a single buffer when demand is uncertain, in a robust framework. We present a family of algorithms based on decomposition that scale well to problems with hundreds of time periods, and theoretical results on more general models. 1
Robust solutions for network design under transportation cost and demand uncertainty
 Journal of the Operations Research Society
"... In many applications, the network design problem faces significant uncertainty in transportation costs and demand, as it can be difficult to estimate current (and future values) of these quantities. In this paper we present a robust optimization based formulation for the network design problem under ..."
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Cited by 8 (0 self)
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In many applications, the network design problem faces significant uncertainty in transportation costs and demand, as it can be difficult to estimate current (and future values) of these quantities. In this paper we present a robust optimization based formulation for the network design problem under transportation cost and demand uncertainty. We show that solving an approximation to this robust formulation of the network design problem can be done efficiently for a network with single origin and destination per commodity and general uncertainty in transportation costs and demand that are independent of each other. For a network with path constraints, we propose an efficient column generation procedure to solve the linear programming relaxation. We also present computational results that show that the approximate robust solution found provides significant savings in the worst case while incurring only minor suboptimality for specific instances of the uncertainty.
Robust and DataDriven Optimization: Modern DecisionMaking Under Uncertainty
, 2006
"... Traditional models of decisionmaking under uncertainty assume perfect information, i.e., accurate values for the system parameters and specific probability distributions for the random variables. However, such precise knowledge is rarely available in practice, and a strategy based on erroneous inp ..."
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Cited by 7 (0 self)
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Traditional models of decisionmaking under uncertainty assume perfect information, i.e., accurate values for the system parameters and specific probability distributions for the random variables. However, such precise knowledge is rarely available in practice, and a strategy based on erroneous inputs might be infeasible or exhibit poor performance when implemented. The purpose of this tutorial is to present a mathematical framework that is wellsuited to the limited information available in reallife problems and captures the decisionmaker’s attitude towards uncertainty; the proposed approach builds upon recent developments in robust and datadriven optimization. In robust optimization, random variables are modeled as uncertain parameters belonging to a convex uncertainty set and the decisionmaker protects the system against the worst case within that set. Datadriven optimization uses observations of the random variables as direct inputs to the mathematical programming problems. The first part of the tutorial describes the robust optimization paradigm in detail in singlestage and multistage problems. In the second part, we address the issue of constructing uncertainty sets using historical realizations of the random variables and investigate the connection between convex sets, in particular polyhedra, and a specific class of risk measures.
Wardrop equilibria with riskaverse users
 Transportation Science
, 2010
"... Network games can be used to model competitive situations in which agents select routes to minimize their cost. Common applications include traffic, telecommunication, and distribution networks. Although traditional network models have assumed that realized costs only depend on congestion, in most a ..."
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Cited by 5 (0 self)
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Network games can be used to model competitive situations in which agents select routes to minimize their cost. Common applications include traffic, telecommunication, and distribution networks. Although traditional network models have assumed that realized costs only depend on congestion, in most applications they also have an uncertain component. We extend the traffic assignment problem first proposed by Wardrop in 1952 by adding random deviations, which are independent of the flow, to the cost functions that model congestion in each arc. We map these uncertainties into a Wardrop equilibrium model with nonadditive path costs. The cost on a path is given by the sum of the congestion on its arcs plus a constant safety margin that represents the agents ’ risk aversion. First, we prove that an equilibrium for this game always exists and is essentially unique. Then, we introduce three specific equilibrium models that fall within this framework: the percentile equilibrium where agents select paths that minimize a specified percentile of the uncertain cost; the addedvariability equilibrium where agents add a multiple of the variability of the cost of each arc to the expected cost; and the robust equilibrium where agents select paths by solving a robust optimization problem that imposes a limit on the number of arcs that can deviate from the mean. The percentile equilibrium is difficult to compute because minimizing a percentile among all paths is computationally hard. Instead, the addedvariability and robust Wardrop equilibria can be computed efficiently in practice: The former reduces to a standard Wardrop
Twostage robust power grid optimization problem
 JNL Operations Research, submitted, 2010. [Online]. Available: http://www.optimizationonline.org/DB FILE/2010/10/2769.pdf
"... For both regulated and deregulated electric power markets, due to the integration of renewable energy generation and uncertain demands, both supply and demand sides of an electric power grid are volatile and under uncertainty. Accordingly, a large amount of spinning reserve is required to maintain t ..."
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Cited by 5 (0 self)
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For both regulated and deregulated electric power markets, due to the integration of renewable energy generation and uncertain demands, both supply and demand sides of an electric power grid are volatile and under uncertainty. Accordingly, a large amount of spinning reserve is required to maintain the reliability of the power grid in traditional approaches. In this paper, we propose a novel twostage robust integer programming model to address the power grid optimization problem under supply and demand uncertainty. In our approach, uncertain problem parameters are assumed to be within a given cardinality or polyhedral uncertainty set. We study cases with and without transmission capacity and ramprate limits. We also analyze solution schemes to solve each problem that include an exact solution approach, and an efficient heuristic approach that provides a tight lower bound for the general robust power grid optimization problem. The final computational experiments on a modified IEEE 118bus system verify the effectiveness of our approaches, as compared to the worstcase scenario generated by the nominal model without considering the uncertainty. Key words: unit commitment, transmission capacity limits, mixed integer programming, separation, robust optimization 1 1